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A somewhat appealing (albeit, to me, also somewhat obscure) view of mathematics is the pluralist doctrine that every consistent mathematical theory is true, insofar as it accurately describes some mathematical structure. I want to comment on a potential worry for this view, mentioned in (Clarke-Doane 2020): that it has implausible consequences for logic.
A famous argument, first proposed in (Lucas 1961), supposedly shows that the human mind has capabilities that go beyond those of any Turing machine. In its basic form, the argument goes like this.
Let S be the set of mathematical sentences that I accept as true. S includes the axioms of Peano Arithmetic. Let S+ be the set of sentences entailed by S. Suppose for reductio that my mind is equivalent to a Turing machine. Then S is computably enumerable, and S+ is a computably axiomatizable extension of Peano Arithmetic. So Gödel's First Incompleteness Theorem applies: there is a true sentence G that is unprovable in S+. By going through Gödel's reasoning, I can see that G is true. So G is in S and thereby in S+. Contradiction!
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